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It is probably because you assign the submit type to your button, your form is submitted before the Firebase method is triggered.
You should change the button code from
<input type="submit" @click="submit" value="Login" class="btn float-right login_btn" />
to
<input type="button" @click="submit" value="Login" class="btn float-right login_btn" />
See the W3 specification for more detail on button types.
Source:stackexchange.com